93

Complex Systems ModelingChristophe Lecerf , Thi Minh Luan Nguyen

christophe.lecerf@ema.fr, ntmluan@ephe-sorbonne.org

Abstract—This paper addresses the simulation of the dynamicsof complex systems by using hierarchical graph and multi-agentsystem. A complex system is composed of numerous interactingparts that can be described recursively. First we summarize thehierarchical aspect of the complex system. We then present a de-scription of hierarchical graph as a data structure for structuralmodeling in parallel with dynamics simulation by agents. Thismethod can be used by physiological modelers, ecological model-ers, etc as well as in other domains that are considered as complexsystems. An example issued from physiology will illustrate thisapproach.

Index Terms—Complex system, hierarchical graph, multi-agentsystem, modeling and simulation

A system is considered as complex when a large number ofelements, arranged in hierarchical structures are involved and,more important, when the interactions take place between ele-ments belonging to different levels (hierarchy, space, time) ofthe organization. For example, information paths and neuronsin the nervous system have each multiple inner levels that wecan model while keeping in sight the interactions between thedifferent levels. The interactions between numerous parts of thesystem produce a emergent behavior that is difficult to predictfrom the collection of elementary specifications. To model thistype of system, we have to take into account not only individ-ual entities features, but also interactions among componentsof different levels that happened in a real system. Traditionalmodeling and simulation methods offer just the vision of macrolevel behaviours, they do not provide insight views of microlevel. Agent-based modeling appears as a suitable tool for sim-ulating complex systems behaviour, in which several thousandof interacting entities can reproduce the real system properties.Modelers can then observe, analyze and study the evolutionof particular components and the emergence of collective be-haviour of the system under study.

In this paper we will study the use of hierarchical graphs formodeling complex systems. More precisely, we will developthe theoretical aspects related to these modeling problems; inparticular we will try to solve the questions related to the map-ping of these hierarchical graphs with the structural organiza-tion of the represented systems on one hand and its dynamicson the other hand.

The paper is organized as follows. Section 1 is dedicated toa short presentation of a complex system, its properties and theneed of simulation in order to observe the evolution of its be-haviour when facing to environmental changes. In section 2,the use of hierarchical graph in modeling the structural organi-

C. Lecerf, Centre LGI2P Ecole des Mines d’Ales - Site EERIE Parc Scien-tifique Georges Besse F 30035 Nimes Cedex 1, France

L. Nguyen, Laboratoire de Recherche en Informatique Avancee, UniversiteParis 8 et EPHE, 41 rue Gay Lussac, 75005 Paris, France

zation is described. In order to simulate the dynamics of bio-logical systems, agent-based simulation is also proposed in thissection. Hierarchical graph play an intermediate role betweenreal complexes systems and its dynamics simulation model. Anexample issued from physiology is described in section 3 inorder to illustrate this approach. We summarize here some im-portant concepts in integrative physiology and the use of ourmodeling method for biological systems. Finally, conclusionsare drawn and future works are outlined in section 4.

We will begin this paper by a brief resume of complex systemand its properties in the next section

I. COMPLEX SYSTEM

A complex system is composed of a set of components, eachof them being itself a set of sub-components, in which variousinteractions between different organization levels take place.Recurrence or hierarchy is the most fundamental characteristicof this kind of system. Their hierarchical aspects are summa-rized in ([?]) by Aiello:

- Abstraction hierarchy- Description hierarchy- Time hierarchy

We have many examples of complex system around us, for ex-ample ecological system, financial market, physiological sys-tem, etc.

One of the most important tasks in studying complex sys-tems is the understanding of their complexity. We would like topresent three techniques summarized by Jennings in [?]:

- Decomposition:it means to divide the main problem inseveral smaller problems. Thanks to the decomposition,a complex system can be divided into many sub-systemsat different level of the structural organization. In the firstanalyze, each of them can be treated as if it was indepen-dent.

- Abstraction: this is the process to define a simplifiedmodel in order to represent the system which emphasizessome details or properties, while suppressing others.

- Organization:this is the process to define and control therelationship among the various components. It allows usto group together certain number of basic components andthen treat them like a single unit of a higher level.

Using these techniques, modeling a complex system becomemore tractable and a hierarchical graph (detailed in the next sec-tion) can be used. It helps us to represent the structure and thecommunication inside these systems and therefore to predicttheir behaviour over time. The common properties of complexsystems allow us to develop a unified resolution method. Byusing this modeling method, the biologists, ecologists, etc will

94

benefit from the results of the parallel research on the complexsystems in other fields than their own.

II. M ODELING WITH DISTRIBUTED COMPUTING

In general, a complex system is viewed as a continuous sys-tem since it receives the activation products from others compo-nents continuously. In order to illustrate this point, we can takethe example of the neuron in the nervous system. Actually, aneuron cell works through the threshold principle, which meansthat it captures all input signals without any macroscopic reac-tion until it reaches its threshold, and then it produces an out-put signal which is a response to the complete series of inputs.While its dynamics is a kind of a step function, at this pointthe discrete event system is more suitable to represent this typeof models. Moreover, distributed system is based on the recep-tion of messages. Thanks to this similarity, we can think aboutthe use of distributed computing techniques in simulation of thephysiological processes.

A. Hierarchical graph

We now introduce briefly the important points of hierarchicalgraph. The detailed definition of hierarchical graph is presentedin [?] by G. Engels, A. Schurr. In short, a hierarchical graph isa graph with two types of nodes: atomic and complex.

• The atomic node is the leaf of the hierarchy and does nothave any internal state.

• On the contrary, the complex nodes have an internal state,which is another hierarchical graph.

A complex node contains a hierarchical graph which may con-tain other complex nodes. In other words, any complex node inthe hierarchical graph is a component of another complex nodeof a higher level. The definition of the hierarchical graphs andthe complex nodes is recurrent. Thus, a unique complex node isused to represent the whole complex system, assumed that eachcomponent is referred to as a complex node: the entire systemis just the highest level one in the hierarchy. We illustrate thispoint in the next figure.

Fig. 1. Hierarchical graph

Hierarchical graph G(N,E)- E: set of edge- N: set of node which is whether atomic node or complex

node.Here we have a simplified description of atomics node:AN =< ID,L >, in which

• ID : node identifier

• L : node’s label.The complex node is described as followCN =<ID,L,C >

• C denoted the internal state which is a complex or atomicnode.

By studying the properties of hierarchical graph, we find outthat there is a perfect coherence between the structural organi-zation of complex systems and the hierarchical graph. It’s re-ally a effective tool for manage hierarchical aspects of the sucha system, for example structural organization level.

The graph is a structure which is used in the modeling of var-ious situations. Actually, a binary relation between objects ofthe same set is enough to define a structure of graph. Therefore,expressed as graphs, many usual problems could be broughtback to traditional problems of the graph theory such as: short-est path; cycle detection; connected components; etc. As weare interested in the dynamics of the model, the graph appearsas a coupling intermediate between the physical system and theassociated mathematical model.

Traditionally, biological processes are modeled mathemati-cally by a set of complicated equations. The traditional modelshave generally intended to offer a global vision of the studyphenomena. Since the biological system involves multiples el-ements that can be modeled in various levels, these equationsdepend not only on global variables but also on local variables.Consequently, this one is not suited for designing behavioursmodels used for simulating a system which involves a greatnumber of interacting elements. This limitation restricts the ef-ficiency of such models and encourages us to use agent-basedmodels for problems where many entities exist and communi-cate with others in some environments.

B. Multi-agent system simulation

The purpose of agent-based simulation is to enlarge our un-derstanding of the multiple processes that occur in biologicalsystems. It consists mainly of representing parts of system ornatural phenomena as a collection of interacting entities thatwork without a direct external influence. In this system, we areinterested in the coordinated behaviour of the individual agentto deduce a system level one.

But what is an agent? According to Wooldridge and Jennings([?]), this is a computer system, situated in some environment,which is capable of flexible autonomous action in order to meetits design objectives. It is defined as active object that:

• Perceptive: agents should perceive their environment andrespond in a timely fashion to changes that occur in it.

• Pro-active: agents should not simply act in response totheir environment; they should be able to exhibit oppor-tunistic, goal-directed behaviour and take the initiativewhere appropriate.

• Social: agents should be able to interact, when appropriatewith others in order to meet its designed objectives.

Since any component of a complex system can be referredas an agent because it has autonomous actions in that environ-ment [?]. Then it possesses all mathematical method to showup a behaviour facing to events which are received from theirentourage environment. Every agent has a set of state. At any

95

Fig. 2. Functional model of the regulation of the respiratory system

time, it is in state S. In the absence of external event, the systemremains on the current state. On the contrary, it receives theexternal event by its input port. The associated mathematicalfunction will then specify how agent change and react due tothis effect. To illustrate this point, we can take an example of aneurone in nervous network. A neurone has a cell body, a set ofinput ports (the dendrites) and a set of output ports (the axons).A neuron has weighted inputs from other neurons and the inputsignals form a weighted sum. Moreover, each neuron has itsown threshold value, if the activation level exceeds the thresh-old, the neuron ”fires” and produces output signals to answer tothe whole series activation.

So, complex system can be seen as a collection of interact-ing entities that work without direct external influence in or-der to meet its objective design. It is a necessary step for au-tomating calculation of its dynamics. Therefore, the hierarchi-cal graph appears as a coupling mean between real complexsystems and the simulation of their dynamics by cooperativeagents. Using hierarchical graph to model the structural orga-

Fig. 3. Relations between a complex system,hierarchical graph and agents

nization of a complex system helps us in the process of defin-ing the agents at work in the model. Actually, exchanges be-tween sub-systems can be modeled by relations between agents.

These functional relations come across the levels defined in thestructural hierarchy.

III. A N EXAMPLE IN BIOLOGY

We now illustrate this approach by presenting an example inphysiology. Consider the following schema (figure 2) that rep-resents the different steps of the regulation of respiratory func-tion: many subsystems are involved, with very different scalesand structures (mechanical subsystems, chemical subsystems,neural subsystems), all interconnected. Modeling such a sys-tem from scratch will a very difficult task.

Hopefully, integrative physiology brings both a conceptualframework and a mathematical tool. The next section is devotedto expose these points

A. Integrative physiology

Traditionally, models in physiology are built independentlyfrom each other, and they are built in an empirical manner.Although the models are being sustained by biological exper-iments and results, different models cannot be easily integratedbecause they do not share any formal theoretical frame.

One could sum up physics as describing the interactions be-tween objects in the physical world with symmetrical forces ina continuous space. Integrative physiology describes the inter-actions between biological entities in a source-to-sink frame,using functional interactions that are non-symmetrical becauseof the structural discontinuity of the biological space. Let usexplain this fundamental point.

Even a simple cell, and moreover a multi-cellular biologicalorganism, is a complex set of substructures in which specificmetabolic process occur. At the scale of a human body, one cansee and feel different organs that make it obvious that the bio-logical space is neither homogeneous nor isotropic. At the very

96

smaller scale of the cell, the space is also divided in local micro-environments, leading to metabolite channelling and poolingphenomena at work in enzymatic metabolism. Since the bio-logical structures interact with each other, whatever its physi-cal scale, the factor supporting the interactions come across thestructural discontinuities, materialized by cell membranes forinstance.

In the integrative physiology conceptual frame, an elemen-tary functional interaction is formally defined by a tripletsource, product, sink and an equation for a field variable (theproduct) driven by a time-space field operator that describes theaction through time and space of the source on the sink. This ac-tion is directed from the source towards the sink, with no feed-back from the sink on the source because the product is trans-formed in the sink. Since the action takes place over a structuraldiscontinuity, whatever its scale, the action of the source on thesink is non-local. For instance, insulin is a hormone (product)produced by specific cells in the pancreas (source), spread di-rectly in the blood flow (structural discontinuity) and used bya muscular cell (sink) to control its glucose inputs. In anothertissue, the example would be one neurone (source) acting on an-other one (sink) by the release of a neurotransmitter (product)in a synapse (structural discontinuity).

Fig. 4. Functional interaction: the general case scheme

In order to describe a biological organism with this concep-tual frame, one has to set up the graph of all the functional inter-actions that occur in the organism. This operation ends up in a(usually large) graph called an organized formal biological sys-tem (O-FBS). The O-FBS should describe both the logical linkof the source on the sink and the geometrical properties of thebiological space in order to make the time-space field operatorusable. Since the speed of the interaction factor (the product)in the biological medium is finite, a functional interaction isnon-symmetrical, non-local and non-instantaneous. It shouldbe noted that this finite speed is the consequence of the specificdynamics of the biological medium (synaptic space, blood, con-junctive tissue, etc.) regarding the specific product considered.

Each functional interaction has its own field variable with itsown dynamics, formalized by an equation summing three termsand referring to source, sink, time and space (figure 4) in the S-propagators formalism. The equation defines the field operatorfor the field variable of the functional interaction, i.e. insulinconcentration or electric potential. The S-propagators formal-

ism was introduced by G.A. Chauvet ([?]) to offer a mathemat-ical representation that provides the integration of elementaryphysiological mechanisms with respect to their structural hier-archy in biological organisms, thus leading to the building ofthe observed physiological functions. Since there also exists afunctional hierarchy in biological organisms, a clear distinctionis made between structural and functional organizations.

Fig. 5. Graph and equation describing the S-Propagator

∂ψ

∂t=D∇2ψ +

∫Dr(r0)

ρr(r′)P (r0)ΦP (r′)ψ(r′, t′)dr′ +

Γr(r0, t)

The first term of the equation describes the local diffusionof the product in the physical space around the source. Thisterm may have a strong influence if the source and the sink arevery close and if the medium can directly transfer the product.For instance, this would be the case for neuronal cells since anaction potential induces a local electric field around it. Con-versely, such a term would have no effect in a long distancehormonal interaction.

The second term is the strictly speaking S-propagator, andrepresents the non-local interaction due to the structural dis-continuity. Therefore, this term describes the hierarchy of theorganizational structure in the physical space. Since the struc-tural discontinuity is also a biological subsystem, it is also de-scribed by a functional interaction with a source, product, sinktriplet and a time-space field operator. This is the most im-portant and most complex part of the S-propagators formalismbecause it makes this theoretical frame homogeneous and stan-dardized with respect to space scale. Most of the biologicalmechanisms are nowadays described by over-simplified mathe-matical models in which the different interactions between sub-systems cannot be taken into account due to the absence of a de-scription with the functional interaction framework. Therefore,these models do not benefit from the S-propagator formalism.

The third term of the time-space field operator describing afunctional interaction represents the source, i.e. the internal lo-cal mechanisms that lead to the generation of the product emit-ted by the source. For instance, in the central nervous system,should the first two terms have put the local membrane potentialabove the local threshold, this term describes the mechanismsgenerating the action potential in the cell.

The S-propagators conceptual framework for describing allthe interactions in a biological system is very powerful as itbrings a standardized formalism and a ”one source - one sink -one product” clue to start the description. Obviously this hintwill help to compare biological and distributed systems.

The main point is that the study of both distributed and phys-iological systems finally goes through simulations in a discrete-event framework: the state of the model changes at only a dis-crete set of simulated time points ([?], [?]). When achieved on

97

a distributed system, such a simulation imposes the network tobe logically synchronized.

Mathematically speaking, though most of the specific math-ematical models that are used have no analytical solutions atthis time (mainly because they are sets of partial differentialequations in which some of the variables are themselves de-scribed by such sets), our assumption stands because the theo-retical frame describing every functional interaction is unifiedand homogeneous.

As shown by Lecerf in his paper [?], there are some com-mon features between distributed computing and physiologicalsystem. We will now summarize these points.

1) A finite state machines network. Both distributed systemsand physiological systems are described as a structuremade of connected components communicating througha point-to-point network. For each component, time isgoing on in one direction and the processes are not re-versible.

2) Finite messages. Units/nodes communicate by means ofmessages, which size is finite. Though this feature hasmuch more meaning for a distributed system than for aphysiological one (it says that a unit cannot spend an in-finite time sending/receiving a message), this assertionstands for both systems.

3) A cost sensitive network. The S-propagators formalismtakes into account the geometry of the modeled biolog-ical system, the length of a space defining the durationof the product propagation. This feature has an equiva-lent in distributed computing with the cost sensitive asyn-chronous networks. In this kind of networks, each linkhas a weight property that is used to measure the commu-nication cost of a particular pathway in the network.

These similarities allow us to apply method presented in sec-tion 2 for physiological system.

B. Hierarchical graph and agent-based method

One should note that such a technological transfer finds ameaning only with the advent of the unified formalism for mod-elling physiological functions made by Chauvet ([?], [?], [?]).Today, graph algorithms and related techniques are obviouslythe common feature between these two poles apart.

The very first step was defining a physiological graph (whichwe called aφ-graph) with special properties for nodes and linksin order to match the needs of integrative physiology model-ing, this data structure allowing us to use some classical graphalgorithms on biological data. For instance, shortest-path algo-rithms can be used on aφ-graph to identify an effective regula-tion loop, thus helping to identify physiological subsystems inthe model O-FBS.

Another application we are working on is to extract from aφ-graph a qualitative description of the regulation loops in order toundergo a formal qualitative analysis of the system’s dynamicsusing the results of R. Thomas ([?]) to find out typical dynamics(like fix points, saddle points, and limit cycles) through a formalmatrix calculus.

The way to represent the biological data (biological entitiesand the physiological processes) on the computer play an es-sential role due to the enormous quantities of information of

such a system. Such an attempt ”cannot be envisaged withoutthe help of powerful computer systems capable of representingand analyzing the intricate networks of physical and functionalinteractions between the different cellular components” [?]. Ingeneral, this complex system forms a large complex networkthat is constructed by a set of biological entities and the inter-action between these entities.

1) Relations between Integrative Physiology and hierarchi-cal graphs: With the directed graph, the unique direction ofthe edges represents the non-symmetry of functional interac-tion. Edges connect nodes of different organization levels; theyrepresent the non-locality of the functional interaction that oc-curs between different structural levels.

The hierarchical level of the graph has a clear correspon-dence with the organization level of the system. As a result,this type of graph can undoubtedly be used for modeling bio-logical systems.

The common and the different features between distributedcomputing and physiological system in particular and com-plex system ([?]) allowing us to take the advantages of dis-tributed system to improve the simulation capacity by distribut-ing agents across computer network.

2) Relations between Integrative Physiology and coopera-tive agents: Both biological and distributed system can be seenas a collection of interacting entities that work without a directexternal control. As shown by Jennings in his paper [?], theagent-oriented philosophy for modeling and managing organi-zational relationships is appropriate for dealing with the depen-dencies and interactions that exist in such a system.

The S-propagator formalism gives us the possibility to simu-late the dynamics of such a system thanks to the homogeneityof the mathematical equations.

Once the structural hierarchy is set, we have to associate eachnode with an agent and give it the adequate methods (describedby S-propagator) in order to describe its activities. They per-ceive changes of the outside environment through input ports.Then they effect to the environment by their output ports. Thecommunication of each component with the external world ismodeled by messages exchange.

Therefore, the functional interaction between the compo-nents of the biological system can now be modeled by relationsbetween agents. When the mathematical expression of the re-lations are tractable, the dynamics of the complex system iscalculable by a computer.

IV. CONCLUSION

Agent-based simulation method does not replace traditionalmethod in biological field. It can be combined with equation-based method in the following way. Within an individual agent,behavioural decisions may be done by the evaluation of equa-tions ([?]). The system level behaviour is then determined byinteractions among these agents.

We presented in this paper the advantages of using hierar-chical graphs for the modeling of the structural organization ofcomplex systems. Associating agents to complex nodes anddescribing functional relations between these agents give a wayto simulate the dynamics of the system. Therefore, hierarchicalgraph, which is an abstract data type ready for programmation,

98

appear as a coupling structure between the real system and thesimulation of its dynamics.

Even with 21st century computers, the simulation of physio-logical systems remains a difficult task, and the S-propagatorsformalism does not seem to facilitate it at first sight since hierar-chical differential partial equations systems exceptionally haveanalytical solutions. But having a unified mathematical frame-work to integrate all the biological data will certainly help tomake more realistic simulations that could be validated by newexperiments. We do not expect any direct application of thiswork in a very short term. Though, should a reliable physiolog-ical model of some subsystems of the human body exist, im-mediate applications would come out: medical care, educationof specialists (physicians, nurses), general education of popula-tions about shock phenomena or cardiac rhythm alterations. Onthe other hand, we hope a feedback for distributed computingthrough the development of new parallel and distributed simu-lation techniques for complex systems. Studying and modelingthe behaviours of a complex system still remains a difficult task.

REFERENCES

[1] Aiello, A. (1997). Environnement Oriente Objet de Modelisation et deSimulationa Evenements Discrets de Systemes Complexes. U.F.R. Sci-ences et Techniques, L’UNIVERSITE DE CORSE: 141.

[2] Jerry Banks (Editor), J. S. C., Barry L. Nelson, David M. Nicol (2000).Discrete-Event System Simulation, Prentice Hall.

[3] Chauvet G.A., (1993a).Hierarchical functional organization of formalbiological systems: a dynamical approach. I. An increase of complexityby self-association increases the domain of stability of a biological sys-tem. Philosophical Transactions of the Royal Society of London SeriesB-Biological Sciences, 339, 425-444.

[4] Chauvet G.A., (1993b). Hierarchical functional organization of for-mal biological systems: a dynamical approach. II.The concept of non-symmetry leads to a criterion of evolution deduced from an optimum prin-ciple of the (O-FBS) sub-system, Philosophical Transactions of the RoyalSociety of London Series B-Biological Sciences, 339, 445-461.

[5] Chauvet G.A., (1993c).Hierarchical functional organization of formalbiological systems: a dynamical approach. III. The concept of non-locality leads to a field theory describing the dynamics at each level oforganization of the (D-FBS) sub-system, Philosophical Transactions ofthe Royal Society of London Series B-Biological Sciences, 339, 463-481.

[6] A.G. Chauvet,Theoretical systems in Biology: Hierarchical and Func-tional Integration. Volume II: Tissues and Organs, Oxford: Pergamon, II, 1996

[7] A.G. Chauvet,Theoretical systems in Biology: Hierarchical and Func-tional Integration. Volume III: Organization and regulation, Oxford:Pergamon, III , 1996

[8] A.G. Chauvet,S-propagators: a formalism for the hierarchical organiza-tion of physiological systems. Application to the nervous and the respira-tory system, Inter Journal of General Systems, 28 (1)53-96, 1999

[9] Chandy K.M., Misra J. (1979).Distributed Simulation: A case study inDesign and Verification of Distributed Programs, IEEE Transactions ofSoftware Engineering, SE-5(5), 440-452.

[10] H. Van Dyke Parunak, R. S., Rick L. Riolo (1998).Agent-Based Model-ing vs. Equation-Based Modeling: A Case Study and Users’ Guide. Pro-ceedings of Multi-agent systems and Agent-based Simulation, Springer.

[11] G. Engels, A.S. (1995). Encapsulated Hierarchical Graphs, GraphTypes, and Meta Types.

[12] Helden J.V., Naim A., Mancuso R.,Eldridge M., Wernisch L., Gilbert D.and Wodak S.J. (2000),Representing and analysing molecular and cel-lular function in the computer, Biol Chem 381(9-10), 921-35.

[13] Jennings, N. R.An Agent-based Approach for Building Complex SoftwareSystems. ACM 44(4): 35-41.

[14] Law A.M., Kelton W.D., (1994).Simulation, modeling and analysis, 3 rded., MacGraw Hill Intl series.

[15] C. Lecerf, M. Bui. (2001). Modeling physiological function: parallelswith distributed processing.Eurosim.

[16] Rickel, J. (1997). Automated modeling of complex systems to answerprediction question. Artificial Intelligence 93: 201-260.

[17] Thomas R., D’Ari R. (1990).Biological Feedback, CRC Press, Boca Ra-ton Florida. Thomas R., (1996).

[18] A. M. Uhrmacher, B. Schattenberg (1998).Agent in discrete event sim-ulation. 10TH European Simulation Symposium ‘Simulation in Industry- Simulation Technology: Science and Art’ (ESS’98), Nottingham, SCSPublications, 129-136.

[19] H. Van Dyke Parunak, R. S., Rick L. Riolo (1998).Agent-Based Modelingvs. Equation-Based Modeling: A Case Study and Users’ Guide. Proceed-ings of Multi-agent systems and Agent-based Simulation, Springer.

[20] M. J. Wooldridge N. R. Jennings. (1995).Intelligent Agents: Theory andPractice. The Knowledge Engineering Review. 10(2). 115-152.